Optical investigations of quantum dot spin ... - APS Link Manager

PHYSICAL REVIEW B 77, 075317 共2008兲

Optical investigations of quantum dot spin dynamics as a function of external electric and magnetic fields Jan Dreiser, Mete Atatüre, Christophe Galland, Tina Müller, Antonio Badolato, and Atac Imamoglu Institute of Quantum Electronics, ETH Zürich, Wolfgang-Pauli-Strasse, CH-8093 Zürich, Switzerland 共Received 24 May 2007; published 19 February 2008兲 We have performed all-optical measurements of spin relaxation in single self-assembled InAs/ GaAs quantum dots 共QDs兲 as a function of static external electric and magnetic fields. To study QD spin dynamics, we measure the degree of resonant absorption which results from a competition between optical spin pumping induced by the resonant laser field and spin relaxation induced by reservoirs. Fundamental interactions that determine spin dynamics in QDs are hyperfine coupling to QD nuclear spin ensembles, spin-phonon coupling, and exchange-type interactions with a nearby Fermi sea of electrons. We show that the strength of spin relaxation generated by the three fundamental interactions can be changed by up to 5 orders of magnitude upon varying the applied electric and magnetic fields. We find that the strength of optical spin pumping that we use to study the spin relaxation is determined predominantly by hyperfine-induced mixing of single-electron spin states at low magnetic fields and heavy-light hole mixing at high magnetic fields. Our measurements allow us to determine the rms value of the hyperfine 共Overhauser兲 field to be ⬃15 mT with an electron g factor of ge = 0.6 and a hole mixing strength of 兩⑀H兩2 = 5 ⫻ 10−4. DOI: 10.1103/PhysRevB.77.075317

PACS number共s兲: 78.67.Hc, 72.25.Rb, 71.35.Pq, 71.70.Jp

I. INTRODUCTION

A single quantum dot 共QD兲 electron spin is a fundamental physical system which allows for a controlled study of confined spin dynamics in the solid state. In contrast to higherdimensional semiconductor structures, QD spins can posses long relaxation and coherence times exceeding 20 ms and 10 ␮s, respectively. The prolongation of spin relaxation times for QD spins stems from a drastic reduction in spinphonon coupling mediated by a combination of electronphonon and spin-orbit interactions, due to strong quantum confinement of electrons. As a consequence, additional spinreservoir interactions such as hyperfine coupling to QD nuclear spins and exchange-type 共cotunneling兲 coupling to a nearby Fermi sea become prominent in determining the spin dynamics in QDs, providing an interestingly rich physical system to study. Major advances in understanding relaxation and coherence of single confined electron spins have been made primarily in electrically defined QDs: Spin lifetimes of ⬃1 ms at high magnetic fields1 共8 T兲 and up to an impressive 170 ms low magnetic fields2 共1.75 T兲 have been recently observed. Rabi oscillations using microwave pulses3 confirmed the long coherence times for electron spins. In coupled QDs, hyperfine-induced singlet-triplet mixing,4 relaxation,5 and coherence6 have been studied. On the self-assembled QD front, optical measurements on InAs/ GaAs self-assembled QD ensembles have revealed T1 times exceeding 20 ms at a magnetic field of 4 T and a temperature of 1 K.7 These findings overall have strengthened the initial proposals for utilizing optical and/or electrical control over QD spins, which act as physical representation of qubits in quantum information processing.8–10 Here, we present a complete experimental and theoretical study of the dynamics of an electron spin confined in a selfassembled InAs/ GaAs QD which is, in turn, embedded in a Schottky heterostructure. In a nutshell, we are able to show 1098-0121/2008/77共7兲/075317共15兲

the interplay of all interactions present in a single QD for the largest parameter range reported to date. We measure the degree of resonant absorption to assess the relative importance and external field dependence of the three elementary spin-relaxation mechanisms, since it is determined by the competition between optical spin pumping11 共OSP兲 and reservoir-induced spin relaxation. Using numerical calculations based on our theoretical model, we are able to obtain an excellent fit to our experimental data. First, we show that at low magnetic fields 共up to 1 T兲, hyperfine interaction shoulders an unexpected dual role, where it alone acts both as a mediator for relaxing 共heating兲 and pumping 共cooling兲 the electron spin in the presence of a resonant laser field. We demonstrate that spontaneous spinflip Raman scattering that allows for one-way pumping into the optically uncoupled spin state is predominantly enabled by a mixing between the electronic spin states induced by the transverse component of the fluctuating nuclear 共Overhauser兲 field. Further, we find that the quasistatic approximation12,13 for nuclear spins is sufficient to explain our experimental results due to the long correlation times of the nuclear spins themselves and to break down the validity of a standard reservoir assumption for nuclear spins. Next, we show that upon varying the external gate voltage by about 50 mV, the spin relaxation due to exchange coupling to the nearby Fermi sea of electrons can be changed by as much as 5 orders of magnitude. Finally, we show that in the high-magnetic-field regime 共2 – 10 T兲, spin pumping is due to heavy-light hole mixing and spin relaxation is dominated by phonons in conjunction with spin-orbit interaction. This paper is organized as follows: In Sec. II, we present a theoretical model that describes the QD spin dynamics in the framework of the trion four-level system with spinreservoir coupling. Experimental results obtained with single QD absorption spectroscopy in distinct regimes of external electric and magnetic fields where different interactions dominate are discussed in Sec. III. Finally, Sec. IV gives an

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FIG. 1. 共Color online兲共a兲 Four-level system describing the singly charged QD in magnetic field along the growth direction 共z axis兲. The electronic ground states with Zeeman splitting ប␻z are vertically coupled by circularly polarized optical transitions to excitonic 共trion兲 states. These consist of a heavy hole and two electrons forming a singlet. The fluctuations of the hyperfine field lead to a slowly varying coherent coupling ⍀H = ⍀H共t兲 of the spin ground states. Incoherent spin-flip processes due to cotunneling and phonon-SO coupling are taken into account by relaxation rate ␬. A laser is introduced at Rabi frequency ⍀R and detuning ⌬␻ from the trion transition. 共b兲 Transformed system after elimination of coherent coupling ⍀H; this system is physically equivalent to that shown in 共a兲. A weak hyperfine-induced diagonal transition appears at rate 2 ˜␥ ⬀ ⍀H / Bz2. The laser is now detuned on the weak ˜␥ transition with a reduced Rabi frequency ⍀R,2.

overview on the above-mentioned interactions together in a self-contained picture. Appendixes are provided at the end for further details and technicalities such as sample structure and experimental techniques for those interested. II. GROUND-STATE OPTICAL TRANSITIONS OF THE SINGLY CHARGED DOT

A singly charged QD is described as a four-level system with two ground states and two excited states, coupled by two vertical optical transitions, as shown in Fig. 1共a兲. The ground state 兩↑典共兩↓典兲 with angular momentum projection mz = + 1 / 2 共mz = −1 / 2兲 is coupled to an excited state 共trion state兲 formed out of two electrons in a singlet and a heavy hole 兩↓↑⇑典共兩↓↑⇓典兲 with spin projection mz = + 3 / 2 共mz = −3 / 2兲. The optical transitions with spontaneous emission rate ⌫ are ␴+ 共␴−兲 polarized according to optical selection rules. The states are defined as

All four states undergo different Zeeman shifts when an external dc magnetic field along the z axis is applied, leading to Zeeman splitting of the optical transitions. A ␴+ polarized laser field is introduced at Rabi frequency ⍀R and detuning ⌬␻ = ␻0 − ␻L, with ␻0 the frequency of the trion transition and ␻L the laser frequency. If only a ␴+ polarized laser field is present, the trion state with mz = −3 / 2, i.e., 共兩↓↑⇓典兲 is inactive since the coupling strength is reduced by a factor exceeding 103 at magnetic fields larger than 60 mT, due to a combination of selection rules and, in the presence of a magnetic field, optical detuning.14 As we shall discuss shortly, the weak spontaneous emission to the other spin ground state cannot be neglected due to its long lifetime. Thus, the system reduces to three levels. The total Hamiltonian describing the system is ˆ ˆ ˆ ˆ ˆ =H ˆ +H H hyp charge + Hphonon + Hint,rad + HZeeman .

The first three terms describe couplings to nuclear spin, freeelectron gas, and phonon reservoirs, respectively. Details about these spin-reservoir couplings can be found in Appendix A. The interaction with the radiation field in semiclassiˆ cal form can be expressed as H int,rad = ប⍀R † i⌬␻t † 共e eQD,−1/2hQD,+3/2 + H.c.兲. The last term describes the Zeeman effect due to an external magnetic field. A. Density matrix description of the quantum dot spin

In what follows, we treat hyperfine interactions via a random quasistatic field and the effect of exchange interactions with the Fermi gas and spin-phonon coupling as dissipative processes with rates described with a Born-Markov approximation. The dynamics of the system is then described using density matrix equations with an effective spin-Hamiltonian and dissipative terms in the Lindblad term. The external magnetic field is aligned with the z axis Bext = Bz = 共0 , 0 , Bz兲. The total magnetic field at the QD is B = Bz + BN, where the nuclear magnetic field 共second term兲 is only seen by the electron spin, but not the hole spin. Taking ˆ the sum of the hyperfine and Zeeman terms yields H hyp ˆ ˆ , with ⍀ 共t兲 as deˆ ˆ +H = ប⍀ 共t兲 ␴ + ប ␻ ␴ + g ␮ B · J Zeeman H x z z h B z z H fined in Eq. 共A4兲. ␴ˆ i are the Pauli matrices, gh is the QD hole g factor, ␮B is the Bohr magneton, and Jˆz is the z component of the hole spin operator. In addition, ប␻z = ge␮B关Bz + BN,z共t兲兴, with ge the QD electron g factor. Using the notation 兩1典 = 兩 ↓ 典, 兩2典 = 兩 ↑ 典, and 兩3典 = 兩 ↓ ↑ ⇑ 典, the unitary part of the Hamiltonian can be written as

冢

† 兩0典, 兩↑典 = eQD,+1/2 † 兩0典, 兩↓典 = eQD,−1/2

† † † eQD,+1/2 hQD,+3/2 兩0典, 兩↓↑ ⇑ 典 = eQD,−1/2 † eQD, ␴

† 共hQD, ␴兲

冣

0 ␻z ⍀H共t兲 ˆ = ប ⍀H共t兲 0 ⍀R . H 0 0 ⍀R ␻0 − ␻L

† 兩0典, 兩 ⇑ 典 = hQD,+3/2

共1兲

is the operator that creates an electron where 共hole兲 in the QD with spin ␴ along the z axis, and 兩0典 is the vacuum 共empty dot兲 state.

共2兲

共3兲

The time evolution of ⍀H共t兲 is much slower than all the time scales over which the system reaches steady state. Therefore, ⍀H共t兲 ⬇ ⍀H is a valid substitution for such a quasistatic system. Each spin-reservoir coupling is included in our three-level model as an incoherent relaxation rate ␬i coupling states 兩1典 ↔ 兩2典 bidirectionally. Each ␬i depends on external mag-

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OPTICAL INVESTIGATIONS OF QUANTUM DOT SPIN…

netic and/or electric field. The coupling of the trion states due to hole spin relaxation has been neglected here due to the low probability of the system being in the excited state manifold. This issue is discussed in Sec. III G. The total spin relaxation rate is

␬ = 兺 ␬i = f共Vg,Bz兲 = ␬cotunnel共Vg,Bz兲 + ␬phonon共Bz兲 + ␬exp . 共4兲 Here, Vg is the gate voltage and ␬exp describes an experimentally induced spin-relaxation rate. Under normal conditions, this relaxation is absent, but can be invoked by largeamplitude gate-voltage modulation in electron cycling experiments as discussed in Sec. III. The explicit three-level Bloch equations are derived in Appendix C for completeness. B. Dressed states and rate equation description of spin pumping

In the following, we will transform the system into another basis that gives an intuitive picture and we will see that optical coupling of the three levels forms a ⌳ system. This allows us to capture the main features of the spin dynamics in the form of rate equations. The new basis states that diagonalize the 2 ⫻ 2 ground ˜ 典 = 兩1 ˜典 spin-state subset of the Hamiltonian of Eq. 共3兲 is 兩↓ ˜ 典 = 兩2 ˜ 典 = sin ␾兩1典 + cos ␾兩2典, and 兩↓ ˜↑ ⇑典 = cos ␾兩1典 − sin ␾兩2典, 兩↑ ˜ = 兩3典 = 兩3典, with ␾ = ⍀H / ␻z. Based on the experimental regimes studied here, we can simplify the calculations since ␾ Ⰶ 1 typically and only take into account first-order terms in ␾. Appendix D addresses this step in more detail. The transformed Hamiltonian then is ˜ =ប H 0

冢

冣

0 ⍀R,1 ␻z 0 0 ⍀R,2 . ⍀R,1 ⍀R,2 ⌬␻

共5兲

˜ = ␾⍀ , ⍀ R,1 R

˜ ␻ = ⌬␻ + ␻ , ⌬ 1 z

共6兲

˜ ␻ = ⌬␻ . ⌬ 2

共7兲

˜ 典 ↔ 兩3 ˜ 典 subsystem: and for the 兩2 ˜⌫ = ⌫,

˜ =⍀ , ⍀ R,2 R

⌫ +2⍀R,2

⍀2 ˜ ⌫ ⍀2 ˜ ⌫ = R,1 . ⬇ 4R,1 2 2 ˜ ␻z2 4␻ +⌫

Under the conditions of ␬ Ⰶ R1→2, R2→1, weak incident beam 共⍀R Ⰶ ˜⌫兲, and a Zeeman splitting largely exceeding the trion decay rate 共␻ Ⰷ ˜⌫兲, we have z

z

˜␳22共t = ⬁兲 ⬇

1 4␻z2 1+ ˜⌫2

共8兲

.

In the case R1→2 Ⰶ ␬, R2→1, ˜␳22共t = ⬁兲 ⬇

1 , 2+␨

␨=

˜␥ ⍀R2 . ␬ ˜⌫2 + 2⍀2

共9兲

R

Hence, in the case of fixed laser intensity, i.e., constant ⍀R2 , the spin-state occupations are determined by the ratio of OSP rate versus spin-relaxation rate. C. Hole mixing

The off-diagonal terms due to ⍀H have been eliminated, and it becomes clear that both ground states couple to the excited state via an optical transition. Also the spontaneous emission terms become modified into a strong and a weak channel, marked by spontaneous emission rates ˜⌫ and ˜␥. The result of the transformation is shown in Fig. 1共b兲: A single laser that interacted with the ␴+ trion transition is now ˜ 典 and 兩3 ˜ 典共兩2 ˜典 represented by two laser field coupling states 兩1 ˜ 典, respectively兲. Effectively, the system can be decomand 兩3 posed into two two-level systems with their own spontaneous emission rates, Rabi frequencies, and effective laser de˜ 典 ↔ 兩3 ˜ 典 subsystem: tunings. Those are for the 兩1 ˜␥ = ␾2⌫,

˜ 典 ↔ 兩3 ˜ 典 subsystem. We start out with the system being in 兩2 ˜ 典. After an intermediate time t given by ˜␥−1 Ⰷ t state 兩2 0 0 −1 Ⰷ ⌫ , the laser field induces a steady-state occupation of the excited state ˜␳33共t0兲. Here, ˜␳ij denotes the element in row i and column j of the transformed systems’ density operator. Further, for times much longer than 关˜␳33共t0兲˜␥兴−1, the system ˜ 典. The net effect of this spin-flip can also be found in state 兩1 ˜ 典 to Raman process is a transfer of occupation from state 兩2 ˜ 典. We will refer to this process as OSP, due to its state 兩1 similarity to experiments performed with atoms.15 Further, we note that a scheme that uses OSP for spin-state preparation had been proposed in Ref. 16 ˜ 典 to state In order to obtain the transfer rate from state 兩2 ˜ 兩1典, and vice versa, under the presence of a resonant laser 共⌬␻ = 0兲, we apply rate equation approximations as discussed ⍀2 in, e.g., Ref. 17 We obtain R2→1 = ˜ 2 R,2 2 ˜␥ and R1→2

In the following, we will discuss the properties of the transformed three-level system with a resonant laser on the

Valence-band mixing, as described by the Luttinger Hamiltonian,18 is a well-known feature in quantum wells. Although being dramatically reduced, it is, nevertheless, expected to play a role in quantum dot dynamics. With valenceband mixing, a heavy hole acquires a small contribution of light holes, and vice versa, such that the effective hole state † as it was defined in Eq. 共1兲 has the form 兩 ⇑ 典hmix = 共hQD,+3/2 † † + ⑀H+hQD,+1/2 + ⑀H−hQD,−1/2兲兩0典 with 兩⑀H⫾兩 Ⰶ 1. Pseudopotential calculations for self-assembled InAs QDs yield admixtures on the order of a few percent.19 As it has been pointed out in Ref. 10, valence-band mixing would have a major impact on the effective optical selection rules by introducing a diagonal relaxation channel between states 兩3典 and 兩1典 due to the admixed light hole component of state 兩3典. Two cases have to be distinguished: First, the mixing contribution associated with ⑀H+ further leads to an effective coherent laser coupling in addition to the coupling induced by hyperfine interaction 共⬀ ⍀␻Hz ⍀R兲 at a detuning ⌬␻ + ␻z as shown in Fig. 1共b兲. Second, the ⑀H− part essentially only appears as a relaxation channel without coherent laser coupling, as the dipole mo-

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ment of this linearly polarized transition lies along the propagation axis of the laser beam and, therefore, cannot be excited. Hence, the following diagonal relaxation terms are ␥ added to Eq. 共C1兲,20 Lˆrelax,hm = 2hm 共2␴ˆ 13␳ˆ ␴ˆ 31 − ␴ˆ 33␳ˆ − ␳ˆ ␴ˆ 33兲 2 2 2 with ␥hm = 兩⑀H兩 ⌫ = 共兩⑀H+兩 + 兩⑀H−兩兲⌫. This diagonal rate leads to OSP in a way similar to the ˜␥ channel enabled by hyperfine interaction. The main difference between hyperfine-induced OSP and valence-band mixing induced OSP is that the first one is magnetic-field dependent as discussed previously, and the latter is not: the valence-band mixing strength ⑀H is expected to be independent of magnetic field as long as the Zeeman splitting is much smaller than the heavy-light hole splitting 共⌬hl ⬎ 10 meV兲, which is true for all realistic experimental magnetic fields. Since the hyperfine-induced OSP rate drops with magnetic field ˜␥ ⬀ Bz−2, hole-mixing-induced OSP should dominate at high fields. From our measurements at high −1 of 2 ⫾ 0.8 ␮s, magnetic fields 关Fig. 7共b兲兴, we extract a ␥hm which yields a hole-mixing strength of 兩⑀H兩 ⬃ 2.2%. This 兩⑀H兩 value is, indeed, much smaller than 1, in agreement with previous theoretical studies. Before closing this section of theoretical considerations, we note that a slightly tilted external magnetic field would yield identical dynamics in the absence of any hole mixing since it would lead to mixing of electronic states induced by the in-plane component of the applied field. These two fundamentally different mechanisms are experimentally indistinguishable for a fixed magnetic-field orientation. Therefore, we repeated our experiments as a function of sample tilt under a magnetic field. For a ⫾1.5° coverage of tilt in all directions, our measurements yielded no observable change in the measured quantity ␥hm. Hence, we can safely state that the inherent hole mixing in our QDs, indeed, dominates over small-angle tilt-induced mixing of electronic spin states. III. SINGLE DOT ABSORPTION SPECTROSCOPY WITH RESONANT LASER A. Experimental method

All data shown in this work have been obtained using differential transmission 共DT兲 technique.21–26 Details regarding our sample and experiment can be found in Appendix B. In order to link the experimentally observable absorption to the three-level system of the singly charged QD, we use the effective ⌳-system picture as described in the previous section. With a resonant laser and large external magnetic field along the z axis, i.e., ⌬␻ = 0, ␻z Ⰷ ˜⌫ Ⰷ ˜␥, the ˜2 ↔ ˜3 subsystem with strong spontaneous emission rate ˜⌫ acts as the main scattering source. The QD response in this type of DT experiments is discussed in the above-mentioned references. In the following, we, therefore, only sketch the link between our three-level system and the intensity of the light transmitted through the sample. The signal detected in a DT experiment arises from interference of the forward scattered field together with the excitation field. This can also be seen from the optical theorem27 which relates the absorption cross section of the dipole to the forward-scattering amplitude. When the QD is

exactly at the focus, the imaginary part of the scattered field is in phase with the excitation laser. After collecting all factors describing spatial mismatch between excitation field and QD scattering cross section, we ˜ 典 ↔ 兩3 ˜ 典 optical define relative absorption caused by the 兩2 transition as ⌰共⌬␻兲 = 1 −

冉

冊

T共⌬␻兲 ⌫ ˜␳32共⬁兲 , = s Im − Toff ⍀R

共10兲

where T共⌬␻兲 refers to the transmitted intensity as a function of laser detuning and Toff to the transmitted intensity far off-resonance in the limit 共⌬␻ → ⬁兲. On resonance in the weak excitation regime, i.e., ⍀R Ⰶ ⌫, the term Im共 ⍀⌫R ˜␳32共⬁兲兲 reaches 1 and s is a scaling factor characterizing the maximum theoretical absorption contrast that is given by s ␴ = Sexp AL0 valid for a weak focusing geometry. Here, ␴0 is the scattering cross section of the two-level system in the weak excitation limit and AL is the laser spot area. The factor Sexp accounts for reduction of signal due to our lock-in detection scheme and experimental imperfections. Further, with ⌬␻ ⌫ = 0, ˜␳32共⬁兲 = −i ⍀R ˜␳33共⬁兲 and Eq. 共10兲 becomes ⌰共⌬␻ = 0兲 = 1 −

T共⌬␻ = 0兲 ⌫2 = s 2 ˜␳33共⬁兲. Toff ⍀R

共11兲

Using Eq. 共11兲, we will infer the value of the spin-up state occupation ˜␳22共⬁兲 from our absorption measurements when varying parameters such as magnetic field and gate voltage but keeping laser power constant, i.e., constant ⍀R. We still need a calibration point, i.e., an experimental value of ⌰共0兲 for a known ˜␳22共⬁兲. In the absence of an external magnetic field, the spin ground states can be considered to be fully mixed due to the in-plane part of the Overhauser field, leading to ˜␥ ⬃ ˜⌫ and a branching ratio of ␩ = 1. As a conse˜ 典 − 兩3 ˜ 典 and the 兩2 ˜ 典 − 兩3 ˜ 典 transitions equally conquence, the 兩1 tribute to light scattering and fast bidirectional OSP takes place, leading to a fully randomized spin, i.e., ˜␳11共t = ⬁ , Bz = 0兲 = ˜␳22共t = ⬁ , Bz = 0兲 = 21 . That having said, expression 共11兲 can be rewritten by introducing a factor s⬘ which can then be experimentally determined ⌰共⌬␻ = 0兲 = 1 −

T共⌬␻ = 0兲 ⌫2 = s⬘ 2 ˜␳22共⬁兲. Toff ⍀R

共12兲

B. Optical spin pumping

Figure 2共a兲 shows absorption on resonance on the blue Zeeman transition as a function of magnetic field normalized to on-resonance absorption at 0 T, i.e., ⌰共Bz兲 / ⌰共Bz = 0兲. The gate voltage was kept in the plateau center, i.e., in a regime where ␬cotunnel is minimal. The inset shows the corresponding raw laser scans for 0 T 共top兲 to 300 mT 共bottom兲. The zero positions of the probe laser detuning has been readjusted in the graphs to compensate the Zeeman splitting. Absorption drops by nearly 2 orders of magnitude over the plotted range of Bz = 0 to 300 mT. With a resonant laser in the weak excitation limit and Zeeman splitting much

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FIG. 2. 共Color online兲共a兲 Absorption maxima in the plateau center plotted as a function of magnetic field Bz. A drop occurs with increasing Bz due to OSP which, at low magnetic fields, dominates over cotunneling and phonon interaction. The gray line is a numerical simulation using ⍀R = 0.6⌫, Bnuc = 15 mT, ⌫−1 = 0.8 ns, addi−1 tional diagonal relaxation ␥hm = 2 ␮s, and ␬−1 = 10 ms. The inset shows the corresponding raw laser scans from 0 T 共top兲 to 300 mT 共bottom兲. The peaks have been shifted laterally to eliminate Zeeman splitting. 共b兲 Optically induced spin pumping rates R2→1, transferring the system into the dark state, and R1→2 the back-pumping rate.

larger than the trion transition linewidth, i.e., ⌬␻ = 0, and 4␻2 ␻z Ⰷ ˜⌫ Ⰷ ˜␥, Eq. 共8兲 yielded ˜␳22共t = ⬁兲 ⬇ 1 / 共1 + ˜ 2z 兲 all pro⌫ vided that the spin-relaxation rate ␬ Ⰶ R2→1 , R1→2, which we can safely assume for low magnetic fields28,29 and strongly suppressed exchange coupling in the gate-voltage plateau center. Consistent with Eq. 共8兲, the drop of absorption follows a B−2 ⬀ ␻z−2 law indicated by the dashed line. For fields less than 100 mT 关see Fig. 2共a兲兴, the approximations included in Eq. 共8兲 do not hold anymore and ˜⌫ ⬃ ˜␥. Without any approximation, the steady-state solutions of the optical Bloch equations are evaluated 共solid line兲 numerically using a fluctuating Overhauser field; they are in excellent agreement with our data at all magnetic fields. The Rabi frequency ⍀R in units of ⌫ for a given incident laser power can be independently determined by saturation spectroscopy and power broadening measurements. The radiative lifetime ⌫−1 = 0.8 ns used in our simulation is based on a measurement in as-grown dots.30 C. Electron cycling

Given that the OSP rates R1→2, R1→2 and the spinrelaxation rate ␬ are unknown, the experimental data shown in Fig. 2共a兲 do not reveal direct quantitative information about ˜␥. However, the branching ratio ␩ can be extracted 2 共t兲典 given in Eq. 共E1兲 using a rms-coherent coupling 具⍀H

␩=

˜␥ ˜⌫ + ˜␥

=

2 具⍀H 共t兲典

␻z2

=

2 Bnuc

2Bz2

.

共13兲

␩ is equivalent to the probability that the system decays via the ˜␥ channel when excited into a trion state.

FIG. 3. 共Color online兲共a兲 Electron recycling measurements. Peak absorption in the plateau center normalized to peak absorption in the cotunneling regime is plotted as a function of laser power at a constant Bz = 300 mT and three different ␬ = ␬exp = 54 kHz 共upper, green points兲, 19 kHz 共middle, red points兲, and 3 kHz 共lower, blue points兲. The fits indicated by the solid gray lines have been obtained using Eq. 共9兲, yielding ␥−1 tot = 0.63 ␮s. 共b兲 A check experiment: Two laser scans at ␬ = ␬exp = 54 kHz with in-and-out of plateau modulation 共showing peak兲 and with in-plateau modulation, demonstrating that, indeed, controlled spin relaxation is realized. The noise level is indicated by the dashed blue line. 共c兲 Intensity dependence of relative absorption at Bz = 0 T 共red circles兲 and Bz = 100 mT 共blue squares兲. There is essentially no dependence on laser power, confirming the theoretical model which gives Eq. 共8兲.

To determine ␩, we applied a large square-wave modulation 共amplitude of 80 mV peak to peak兲 at different frequencies to the gate which, in every cycle, first loaded another electron of opposite spin into the QD, forming a singlet together with the QD electron. Then one of the electrons was forced to leave, and as the tunneling probability for each of the two electrons is equal, the remaining QD spin was fully randomized. The advantage of this technique, which we will refer to as electron cycling, leads to enforced spin relaxation at a known and controlled rate ␬exp of Eq. 共4兲. In the case ␬ ⬇ ␬exp Ⰷ R1→2, i.e., enforced spin-relaxation rate exceeds the optical back-pumping rate, R2→1 and ˜␥ can be determined by a fit using Eq. 共9兲. Figure 3共a兲 shows plateau-center absorption normalized to on-resonance absorption in the cotunneling regime; the data were obtained with electron cycling for different modulation frequencies and laser powers at a fixed external magnetic field Bz = 300 mT. The upper, green points correspond to ␬ = 54.3 kHz, the middle, red point to ␬ = 19.3 kHz, and the lower, blue points to ␬ = 3.3 kHz. Using Eqs. 共12兲 and 共9兲, the absorption ratio shown in the figure can be written as 2 ␪in-plateau ␳22,in-plateau = = . ␪cotunnel ␳22,cotunnel 2 + ␨

共14兲

The gray lines are fits using this expression. Best match with the data can be obtained with a total OSP rate ␥tot共300 mT兲 = ˜␥共300 mT兲 + ␥hm = 1.6 ␮s−1,31 where the two contributions stem from nuclear spins and hole mixing, re-

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FIG. 4. 共Color online兲 Absorption strength as a function of laser detuning and gate voltage. External magnetic field of Bz = 150 mT is applied. The linear gate-voltage dependence is due to the quantumconfined Stark effect. 共a兲 and 共c兲 show the absorption in the vicinity of the blue optical transition, 共b兲 and 共d兲 show the red transition. 共a兲 and 共b兲 are experimental data showing a spectral shift of the weak absorption peak in the plateau center compared to the strong cotunneling regimes at the plateau edges. The shift is directed to the lower energies for the blue transition, and toward higher energies for the red transition. This feature can be reproduced in a numerical simulation including a randomly fluctuating Overhauser field as shown in 共c兲 and 共d兲 using the parameters ⍀R = 0.6⌫, ⌫−1 = 0.8 ns, −1 Bnuc = 15 mT, and ␥hm = 2 ␮s.

spectively. The hole mixing-induced contribution can be independently determined from high-magnetic-field measure−1 = 2 ␮s. Using the branching ratio 共13兲 with ments to be ␥hm ˜⌫−1 = 0.8 ns, we then solve for the rms-nuclear magnetic field and obtain Bnuc = 15mT, which is in good agreement with the number obtained in Appendix A 1. Figure 3共b兲 shows a measurement that demonstrates the difference between electron cycling 共large-amplitude modulation兲 and in-plateau 共small amplitude兲 modulation: For inplateau modulation, we do not observe absorption 共the noise level is marked by the horizontal dashed line兲. In contrast, when large-amplitude modulation is applied, absorption is partially recovered due to forced spin relaxation at a controlled rate ␬exp, as shown by the red peak in the figure. Figure 3共c兲 was obtained in the plateau center without electron cycling technique, showing on-resonance absorption as a function of incident laser power. At 100 mT 共blue squares兲, relative absorption is 1 order of magnitude weaker than at 0 mT 共red circles兲 due to OSP. In both cases, absorption exhibits weak dependence on laser power. This dependence arises from saturation at high power levels, while the experimental parameters extracted via this method are from the power regime of Fig. 3共a兲 well below saturation.

FIG. 5. 共Color online兲共a兲 Trion transition laser scans for five different magnetic fields. The gate voltage was in the cotunneling regime 共see Appendix A 2兲. 共b兲 Measured linewidth obtained from laser scans as a function of magnetic field 共indicated by the red circles兲. A broadening occurs at Bz ⬇ 75 mT; at larger fields, the linewidth almost recovers back its original value of 450 MHz at 0 T. The solid line is obtained via numerical simulation with pa−1 rameters ⍀R = 0.6⌫, Bnuc = 15 mT, ⌫−1 = 0.8 ns, ␥hm = 2.0 ␮s, and −1 ␬ = 2.5 ␮s.

confined Stark effect and the abscissa pixelization is due to gate-voltage steps during each scan. At gate voltages of 395 and 480 mV, the two cotunneling regimes show strong absorption when spin relaxation is fast due to charge reservoir coupling 共for details, refer to Appendix A兲. In the plateau center, hyperfine interaction dominates and leads to spin pumping and drop of absorption as already discussed. Here, we further observe a shift of the spectral position of the absorption peak in the pleateau center as compared to the cotunneling regime. This shift is directed to the red 共blue兲 for the blue 共red兲 Zeeman transition. This resembles effects one might expect for dynamical nuclear spin polarization32,33 共DNSP兲, these effects can, however, be excluded.34 Our numerical simulation is able to reproduce this behavior without taking into account DNSP, as shown in Figs. 4共c兲 and 4共d兲. The line shift in the plateau center is ⫾0.9 GHz for both red and blue transitions, which is close to the electronic Zeeman splitting at 150 mT, EZ,e ⬇ 1.3 GHz. At the cotunneling edges, ␬ is large and the maximum of absorption is observed when the laser is exactly on-resonance with the transition. When ␬ is small, as it is the case in the plateau center, absorption of a strictly resonant laser is suppressed due to spin pumping, depending on the external magnetic field. When the laser frequency is moved toward the center between the strong ˜⌫ and the weak ˜␥ transitions, i.e., the spectral detuning with respect to the ˜␥ transition is reduced, the backpumping at rate R1→2 becomes more efficient, and maximum of absorption will be reached for a spectral detuning that fulfills the condition R1→2 = R2→1. As a consequence, both transitions contribute to absorption, which leads to a shift of the absorption maximum toward the weak ˜␥ transition, i.e., a blueshift when the red line is observed and vice versa.

D. Peak shift in the plateau center

Figure 4 shows laser scans obtained at Bz = 150 mT on the blue 关Fig. 4共a兲兴 and red 关Fig. 4共b兲兴 Zeeman transition throughout the whole single-electron plateau. Absorption strength is color coded. The line tilt is due to the quantum-

E. Peak broadening at plateau edges

Figure 5共a兲 shows example laser scans for different magnetic fields Bz ranging from 0 to 1 T, obtained in the cotunneling regime where spin relaxation is fast. The scans have

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F. Coupling to electron spin reservoir

FIG. 6. 共Color online兲共a兲 Example voltage coarse scan across the entire single-electron plateau at Bz = 300 mT. Per voltage step, a laser scan is performed and the observed absorption maximum is plotted as a function of gate voltage. Due to strong hyperfineinduced OSP and weak cotunneling, rate absorption in the plateau center is suppressed; however, when approaching the singleelectron plateau edges, absorption is recovered due to highly nonlinear dependence of cotunneling on gate voltage, leading to fast spin flips. Outside the voltage plateau, i.e., left of point A or right of point B, absorption is suppressed because the QD then becomes either empty or doubly charged, which shifts the optical transition energies out of our spectral observation window of 30 GHz. The solid line is a guide for the eye. 共b兲 shows a voltage fine scan of the left cotunneling regime obtained on another QD than in 共a兲. The solid line is a numerical calculation using ⌫−1 = 0.8 ns, ⍀R = 0.6⌫, −1 Bnuc = 15 mT, ⌫−1 tunnel = 20 ns, ␥hm = 2 ␮s, and electronic Zeeman splitting EZ,e = 10 ␮eV. The voltage full width at half maximum 共FWHM兲 of the cotunneling peak is 10 mV.

been laterally shifted in order to eliminate the Zeeman shift. In Fig. 5共b兲, the measured linewidths are plotted as a function of magnetic field 共red circles兲 along with a calculated curve 共solid line兲. A broadening to almost double the zerofield linewidth appears at magnetic fields between 60 and 80 mT; at higher fields, linewidth becomes as narrow as in the case Bz = 0. The physical reason for the observed broadening is very similar to that described in Sec. III D: Both ˜␥ and ˜⌫ transitions contribute in a non-negligible way to absorption, and a maximum is observed when R1→2 = R2→1 condition is fulfilled. Consistently, the linewidth increases as much as the electronic Zeeman splitting initially, but drops at magnetic fields where ˜⌫共B兲 Ⰷ ˜␥共B兲 and a single transition is established. In contrast to Sec. III D, ␬ is large here due to cotunneling and annihilates the absorption drop caused by spin pumping, hence making the transition visible at all magnetic fields. The solid line is a calculated curve using a randomly fluctuating Overhauser field with Bnuc = 15 mT, well reproducing this feature. We note that in order to put as many constraints as possible on the choice of simulation parameters, we have used the maximum cotunneling-induced spinrelaxation rate ␬ = ␬cotunnel = 0.4 ␮s−1 as obtained from the data shown in Fig. 6共b兲. We find that the effect of ␥hm, i.e., ⑀H, on the simulation is negligible, advocating that the dominant OSP mechanism is hyperfine interaction.

We have performed laser scans as a function of gate voltage througout the whole single-electron plateau as defined in Appendix A. The measured on-resonance absorption signal for each laser scan is plotted in Fig. 6共a兲. The data have been obtained at an external magnetic field of Bz = 300 mT taking coarse voltage steps. At gate voltages lower than 540 mV and higher than 625 mV, as marked by the shaded regions, absorption drops below noise level indicated by the horizontal dashed line. At these voltages, the QD either becomes empty 共left of point A兲 or doubly charged 共right of point B兲. Absorption then vanishes since, in those cases, the QD is not described by the trion level system anymore; the optical transitions for these gate voltages are not observed within our scanning window of 30 GHz around the trion transitions. The unshaded part indicates the region where the QD contains a single electron and, as it has been mentioned before, the cotunneling rate is maximum when gate voltage is at the crossover points A or B. Here, relaxation via cotunneling is faster than the optical pumping rates ␬cotunnel Ⰷ R1→2 , R2→1, leading to thermalization of the electron spin and, thus, strong absorption. The scenario drastically changes when gate voltage is tuned to the center of the plateau. Here, co-tunneling rate ␬cotunnel reaches its minimum, where our numerical calculation predicts a drop of as much as 5 orders of magnitude 共also see Fig. 10兲 compared to the crossover points such that ␬cotunnel Ⰶ R2→1. Consequently, the occupation of the spin states is governed by OSP 关Eq. 共8兲兴 rather than Boltzmann factor, meaning that the spin is predominantly in the dark state and vanishing absorption is observed. The semilogarithmic plot in Fig. 6共b兲 shows a voltage fine scan of the low voltage plateau edge around the A crossover point obtained at Bz = 300 mT. The gate voltage for point A is different from Fig. 6共a兲 as these data were taken on another QD. The observed absorption drops by half within a gate voltage detuning of ⫾5 mV from the maximum position. These data demonstrate a giant gate-voltage dependence of this spin-relaxation mechanism. The gray solid line is a bestfit numerical simulation using expression 共A6兲 as spinrelaxation rate, showing good accordance with the data. The cotunneling rate at the peak as determined from the fit is −1 ␬max = 2.5 ␮s. The noise level is indicated by the dashed line; it deviates from the one shown in Fig. 6共a兲 due to different experimental settings such as lock-in time constants and filter slopes. Again, on the left side of the peak, the QD is empty, yielding vanishing absorption below the noise level. The gradual decrease of absorption is due to finite temperature. On the right side, the spin pumping regime is located; here, some weak absorption remains according to the occupation of the observed spin state, revealing the strength of spin pumping. G. Coupling to phonon reservoir

Based on the theoretical estimates of Appendix A, we now seek for signatures of the last remaining reservoirinduced spin dynamics, i.e., spin-orbit 共SO兲-phonon assisted spin relaxation at high magnetic fields. Figure 7共a兲 shows

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FIG. 7. 共Color online兲共a兲 Plateau laser scans for four different magnetic fields: Absorption is plotted as a function of laser detuning and gate voltage. Zeeman effect has been eliminated by vertical shift of each single plot. The linear voltage dependence is due to quantum-confined Stark effect. Plateau-center absorption drops at intermediate magnetic fields due to OSP, but then recovers at high fields due to fast thermalization via phonon-SO interaction at 9.9 T. At the plateau edges, the QD can absorb at all magnetic fields due to fast ␬cotunnel. 共b兲 Black data points: Magnetic-field evolution of peak absorption in the plateau center normalized to peak absorption in the cotunneling regime. The lower 共upper兲 boundary of the gray −1 region is obtained from a numerical simulation for ␥hm = 1.2 ␮s −1 −1 共␥hm = 2.8 ␮s兲; the red line corresponds to ␥hm = 2 ␮s. Best match with the data yields explicitly for the spin-relaxation rate ␬phonon = ␣0Bz5 with ␣0 = 0.031 in units of T−5 s−1. Again, Bnuc = 15 mT.

two-dimensional 共2D兲 plots of color-coded absorption strength as a function of laser detuning and gate voltage for four different magnetic fields obtained for the red Zeeman transition. The scans cover the whole single-electron plateau; excitonic Zeeman shift has been eliminated by shifting the y scale for each 2D graph separately. The linear dependence of the excitonic transition energy on gate voltage is due to the quantum-confined stark shift. At 0 T, absorption is clearly visible throughout the whole plateau due to fast spin flips with the neighboring nuclear spins. When a small magnetic field 共B = 0.1 T兲 is applied, absorption in the plateau center drops because of hyperfineinduced OSP, as discussed in the previous sections. Close to the plateau edges, absorption still remains due to fast cotunneling. At 0.5 T, increasing OSP leads to further drop of absorption. These absorption characteristics in the plateau center remain the same up to 5 T; however, absorption starts to come back at even higher fields: when the magnetic field is raised up to 9.9 T, a significant recovery of plateau-center absorption is observed. This effect cannot be explained by OSP, which only causes monotonous decrease of absorption, nor by cotunneling, which is negligible in the plateau center and hardly shows any magnetic-field dependence. Owing to its Bz5 dependence, however, phonon-assisted spin relaxation is a good candidate for the origin of the observed effects in the context of spin-relaxation mechanisms. Figure 7共b兲 shows the quantitative evolution of normalized absorption with magnetic field, i.e., the ratio of absorp-

tion in the plateau center versus cotunneling regime 共black data points兲. Further, the solid red line along with the gray shaded region indicates the calculated strength of absorption −1 for ␥hm = 2 ␮s with an uncertainty of ⫾0.8 ␮s; the phononinduced spin-relaxation rate was ␬phonon = ␣0Bz5, with the coefficient ␣0 = 0.031 in units of T−5 s−1. Whereas ␬phonon is strongly B dependent, the hole-mixing contribution ␥hm has no B dependence within the magnetic-field range considered here. Therefore, these two mechanisms have distinguishable effects on Fig. 7共b兲 and, thus, can be identified independently. The good agreement with the experimental data strongly suggests that in this regime of electric and magnetic fields, the dominant spin relaxation is, indeed, phonon assisted. Further, within our uncertainty, ␬phonon matches well with the results that have been previously obtained on an ensemble of self-assembled InAs/ GaAs QDs.7 There are two fundamentally different mechanisms which employ holes to yield OSP: First, hole mixing of strength ⑀H leads to an admixture of the light hole states to the trion states as dicussed in Sec. II C. Second, hole-spin relaxation leads to an incoherent coupling of the trion states contributing to OSP. In Ref. 35, hole spin-relaxation rate is predicted to be below 103 / s and monotonically increases with magnetic field, which suggests that it is not the main mechanism responsible for OSP. We, therefore, neglect hole spin flips, further assuming that there is no other efficient hole spin-flip mechanism at low magnetic fields. In the first mechanism, OSP is independent of magnetic field and the strength is equal to the hyperfine-induced OSP rate at ⬃1 T. At higher magnetic fields, the hyperfine-induced OSP rate drops with Bz−2; therefore, hole mixing becomes the dominant OSP mechanism here. IV. FULL INTERACTION MAP

In the previous sections, three spin-relaxation mechanisms acting on the confined spin have been identified separately along with the two mechanisms for OSP, both through experimental and theoretical studies. In this final part, we extrapolate our findings numerically to the whole of the relevant external magnetic and electric field phase space in order to predict the longest available single-electron spinrelaxation times within this whole range. The calculations have been performed within a parameter space approximately overlapping with the full scale of our experimental tuning ability of the static electric and magnetic fields. For details of the simulation, we refer to Appendix E. Figure 8 shows calculated maximum values of absorption for the red 关Fig. 8共a兲兴 and the blue 关Fig. 8共b兲兴 trion transition with a laser having the corresponding circular polarization. Absorption strength is color coded in logarithmic scale as a function of gate-voltage detuning and external magnetic field. All of the following points have been discussed in the previous sections; here, we mention them briefly as a key to the plots: the necessary conditions for observing strong absorption are either ␬ Ⰷ R1→2 , R2→1 or R1→2 ⬃ R2→1. Further, at large magnetic fields Bz ⬎ 8 T when electronic Zeeman splitting EZ,e ⬃ kT, the Boltzmann factor leads to a difference

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FIG. 8. 共Color online兲 Calculated absorption maxima for the whole single-electron plateau plotted as a function of magnetic and electric fields. 共a兲 shows the simulation for the probe laser in the vicinity of the red Zeeman transition; 共b兲 similar but for the blue Zeeman transition. The borders of each plot show strong absorption due to interactions with nuclear spins 共left兲, charge reservoir 共top/ bottom兲, and phonon reservoir 共right兲. At large magnetic fields, spin polarization nearly reaches unity due to thermalization, leading to vanishing absorption on the red transition 共a兲 and enhanced absorption on the blue transition 共b兲. In the center of the plots, absorption, and thus spin relaxation, is suppressed by approximately 5 orders of magnitude. The parameters used in the simulation are ⌫−1 = 0.8 ns, −1 Bnuc = 15 mT, ␥hm = 2 ␮s, tunneling time ⌫−1 tunnel = 20 ns, and ␬phonon as given in Sec. III G.

of the spin ground-state occupations and, thus, a difference between absorption strengths on the red and blue Zeeman transitions. In the plot, we distinguish three different regimes of strong absorption: 共1兲 Magnetic fields lower than the fluctuations of the hyperfine field 共Bz ⱗ 15 mT兲. Here, fast bidirectional OSP due to hyperfine-induced state mixing leads to strong absorption. 共2兲 High magnetic fields 共⬎5 T兲. Here, ␬phonon induces fast thermalization, i.e., ␬phonon Ⰷ R1→2 , R2→1. The spin ground-state occupation is mainly determined by the Boltzmann factor, leading to a lowering 共increase兲 of absorption on the higher 共lower兲 energy spin state occupation 共a兲关共b兲兴. 共3兲 Large gate-voltage detunings from the plateau center 共⫾40 mV兲. Here, cotunneling 共␬cotunnel兲 is responsible for fast spin relaxation and appearance of absorption. An intriguing feature that becomes apparent now is the blue island in the center of the color-coded spin-relaxation plot. It marks the regime where absorption 共i.e., all reservoir interactions兲 is suppressed by 5 orders of magnitude or, in other words, the localized spin becomes maximally isolated, hence the frequently used concept of an artificial atom is meaningful. Within the scope of quantum information processing, this indicates the relevant regime of operation where a spin-relaxation time of up to 1 s is predicted.

coupling becomes important. For magnetic fields Bz ⱗ 1 T, the dominant contribution to OSP stems from the fluctuating hyperfine field mixing the electronic spin states and creating a weak channel for diagonal relaxation in the trion four-level picture. Exchange and phonon-induced spin-flip processes dominate over hyperfine-induced spin pumping and establish a thermal steady state at large gate-voltage detunings and/or large external magnetic fields; in the plateau center at intermediate magnetic fields, the situation is reversed and spin pumping dominates, strongly altering the state occupations away from thermal equilibrium values. Signatures of heavylight hole mixing dominated spin cooling can be observed for fields ⲏ5 T. From a quantum control perspective, these results demonstrate that the quantum dynamics of a single confined spin can be significantly altered by externally controlled parameters such as electric and magnetic fields. A natural extension of this study would be the investigation of spin decoherence in a single QD using similar optical techniques. These measurements would require more advanced schemes such as electromagnetically induced transparency. Further, knowledge gained on single-electron spin dynamics can be utilized in the resonant optical study of more complex systems such as coupled QDs or QDs with a single excess heavy hole. ACKNOWLEDGMENTS

We thank Hakan Türeci, Alex Högele, Tunc Yilmaz, Jeroen Elzerman, and Nick Vamivakas for useful discussions. This work is supported by NCCR Quantum Photonics. J.D. and M.A. would like to thank J. Cash for technical assistance. APPENDIX A: RESERVOIR COUPLINGS OF THE QUANTUM DOT SPIN 1. Nuclear spins

The interaction of a localized electron spin with a surrounding nuclear spin ensemble can be written in the form of ˆ H the Fermi contact interaction12,13,29,36 hyp ␯0 ˆ 兲. The sum runs over all nuclei i in the = 8 兺iAi兩␺共Ri兲兩2共Iˆ i · ␴ lattice. ␯0 is the volume of an InAs unit cell, ␺共Ri兲 the elecˆ tron envelope wave function at the ith nucleus, and Iˆ i and ␴ are the spin operators of nuclear and electron spins. Ai is the hyperfine coupling strength determined by the value of the electron Bloch wave function at the site of each nucleus. The total number of nuclei in our InAs/ GaAs QDs can be estimated to be N = 104 – 105. We can equivalently describe the effect of hyperfine interactions with an effective magnetic field seen by the QD spin, which is commonly referred to as Overhauser field BN ␯ ¯ = 0 A 具兺 Iˆ 典, where ¯A is an average spin-nuclei coupling 8 g e␮ B

V. SUMMARY AND CONCLUSION

We have investigated the dominant interactions of a confined electron spin in a single self-assembled QD by optical means and demonstrated the regimes where each reservoir

i i

constant. Due to the arbitrary direction of the Overhauser field, the spin ground states become admixed. The excited 共excitonic兲 states remain unaffected due to the p-like symmetry of the hole Bloch wave function and to the two elec-

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FIG. 9. 共Color online兲共a兲 Stability diagram of the QD ground states neglecting spin: Energies of the zero, one-, and two-electron QD as a function of gate voltage. Crossover points are marked A and B. E12 denotes the gate-voltage-dependent energy difference between the singly charged and doubly charged states or charging energy. 关共b兲–共d兲兴 The QD can exchange its single electron with the charge reservoir via a virtual empty or two-electron 共shown here兲 state. When one of the two singlet electrons tunnels out, it leaves the remaining QD spin in a mixed state equivalent to spin relaxation. ⌫tunnel marks the tunneling rate through the 35 nm GaAs barrier, and ␧ the detuning from the Fermi energy ␧F. Ez is the electronic Zeeman splitting, and E12 the energy required to charge a second electron.

trons forming a singlet which is immune to magnetic-field variations. In order to further understand the effect of the hyperfine field in optical experiments, we consider two regimes: First, an external magnetic field with strength smaller than the hyperfine field is applied or the external field is completely absent. Hence, the direction of the total magnetic field seen by the electron spin is fully random after a nuclear field correlation time. As we saw in Sec. II, fast bidirectional OSP is the consequence of inducing efficient spin relaxation to dominate over other mechanisms.29 In the second regime, the applied external field is much stronger than the hyperfine field. In this case, the electron spin mainly sees the external magnetic field along the z axis and the hyperfine field only leads to small fluctuations of the nuclear field vector. In this regime, the light-induced spin relaxation is slow and other mechanisms can be dominant. For the experiments described in this paper, we can treat the hyperfine field as a purely classical field BN共t兲 with correlation time ␶corr ⬃ 1 ms. The correlation time is expected to be similar to the decay time of nuclear spin polarization in the presence of a QD electron and in the absence of external magnetic field, as measured in Ref. 37. Bnuc refers to the rms value of the Gaussian distribution as defined by f共BN兲 =

1 3 Bnuc 共2␲兲3/2

冉

exp −

兩BN兩2 2 2Bnuc

冊

,

共A1兲

2 . Here, 具典 dewhich yields 具BN共t兲典 = 0 and 具兩BN共t兲兩2典 = 3Bnuc notes the time average over many correlation times. Bnuc can be written in the form

Bnuc =

b0

冑N ,

共A2兲

with b0 a parameter characterized by the species of nuclei and the composition of the QD,38 and N the number of nuclear spins interacting with the QD spin. The QD composition is taken to be 90% InAs and 10% GaAs, yielding I共I + 1兲 = 13.2 when averaging over the different nuclear species.39 Similarly, we obtain A2 = 2500 ␮eV2, which yields b0 = 3.0 T. Using Eq. 共A2兲 with N = 104 – 105 nuclear spins, we obtain for our QDs Bnuc = 9.5– 30 mT. As our BN共t兲 is classical, we treat the BN,i共t兲 with i = x, y, and z

as independent random variables. Here, the component of the nuclear field along the z axis BN,z共t兲 only leads to Zeeman splitting, whereas the in-plane components induce a mixing of the 兩↑典 and 兩↓典 states. The in-plane hyperfine field is 2 共t兲 + B2N,y共t兲, B2N,xy共t兲 = BN,x

共A3兲

and we define ប⍀H共t兲 =

ge␮BBN,xy共t兲 . 2

共A4兲

We note here that our measurement time 共typically 10– 100 ms兲 is longer than the correlation time of the nuclear field; i.e., for each measured data point, we expect that we average over many configurations of the nuclear magnetic field.40

2. Coupling to electron spin reservoir

The exchange interaction with the Fermi sea in the back contact 共sample details are in Appendix B兲 can be written as † † ˆ H charge = 兺 បgt,k共ek,↓eQD,↑eQD,↓ek⬘,↑ + c.c.兲,

共A5兲

k,k⬘

† where eQD, ␴ and eQD,␴ are the creation and annihilation operators for an electron with spin ␴ in the QD, and similarly in the reservoir. gt,k is the tunneling matrix element, which is linked to the tunneling rate ⌫tunnel by Fermi’s golden rule ⌫tunnel = 2ប␲ 兩gt,k兩2␳共E兲, with ␳共E兲 being the density of states in the back contact. It is well known that exchange interaction of a confined spin with an electron spin reservoir gives rise to spin-flip cotunneling41,42 at our operating temperatures 共T ⬃ 4 K兲; at temperatures lower than the Kondo temperature TK, it leads to the formation of a Kondo singlet.43–46 Figure 9共a兲 shows the energies of the empty, singly, and doubly charged QD state as a function of gate voltage.47 Which state has the lowest energy obviously depends on the gate voltage, and the QD attempts to reach it by either attracting or repelling electrons from or into the reservoir. Clearly, there is a range of voltages 共single-electron charging plateau兲 where it is energetically favorable for the QD to accomodate a single electron, marked by the shaded region

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in the figure. At the points A and B, two charging levels are degenerate and fast exchange of the QD electron with the reservoir can take place, only limited by the tunneling rate. The real gate voltages VA and VB that need to be applied in order to reach points A and B can vary from dot to dot, depending on its confinement properties. We define the plaV −V teau center Vc = B 2 A . The gate-voltage detuning is ⌬Vg = Vg − Vc. The schematic cotunneling process is depicted in Figs. 9共b兲–9共d兲. The initial state is characterized by a QD with a single spin-down electron, and Coulomb blockade prohibits tunneling of further electrons into the dot 关Fig. 9共b兲兴. Together with a spin-up electron from the reservoir, a virtual spin singlet 关Fig. 9共c兲兴 is formed at energy difference ⌬E = ␧ + E12, where ␧ is the detuning from the reservoirs’ Fermi energy ␧F. Finally, the QD returns to the singly charged state with a spin-up electron 关Fig. 9共d兲兴. E12 is given by E12 = E2 − E1 = e 共Vg−VA兲

共VB−Vg兲 ␭

and E01 = E1 − E0

= e ␭ , with Ei the energy of the QD charged with i electrons and ␭ a constant describing the geometric lever arm of the heterostructure. Using Eq. 共A5兲, one obtains for the cotunneling rate in second order42,48 2 ␬cotunnel = ប⌫tunnel

+

冕

␧

冨

1 e共Vg − VA兲 i + ␧ + ប⌫tunnel ␭ 2 1

e共VB − Vg兲 i − ␧ + ប⌫tunnel ␭ 2

冨

2

FIG. 10. Expected cotunneling rate obtained using expression 共A6兲 with the parameters ⌫tunnel = 0.1 ns−1 共solid curve兲 and ⌫tunnel = 0.02 ns−1 共dashed curve兲, VA = −50 mV, VB = + 50 mV, kT = 300 ␮eV, and ␭ = 5.3.

pression 共A6兲 using two different tunneling rates of ⌫tunnel = 0.02 ns−1 and ⌫tunnel = 0.1 ns−1 representing the minimum and the maximum cotunneling rate we expect in our experiments respectively. Cotunneling rate is characterized by its very nonlinear voltage dependence. When close to the crossover points VA and VB, it exhibits an ultrasteep slope; in contrast, the voltage dependence is weak in the plateau center Vc.

f共␧兲关1 − f共␧兲兴d␧. 3. Spin-phonon interaction

共A6兲

The integral is the sum over all second-order transitions with different detunings ␧ from the Fermi energy according to Figs. 9共b兲–9共d兲. In addition, the term with e共Vg − VA兲 / ␭ = E01 describes the related process where the virtual state is an empty QD. f共␧兲 is the Fermi function f共␧兲 = 1 / 关1 + exp共␧ / kT兲兴. Expression 共A6兲 is valid under the condition EZ,e Ⰶ kT, i.e., for low magnetic fields. To obtain the exact expression for all magnetic fields, the Fermi function terms in the integral have to be modified.49 The imaginary part of the denominator introduces a finite lifetime to the electronic states limited by the tunneling rate ⌫tunnel, implying that the main cause for broadening of the spin ground states is tunneling. This is relevant for elements of the integral with vanishing real part. In order to obtain an estimate for the cotunneling times in our structure, we use results obtained on samples with 25 nm tunneling barrier where in certain gate-voltage regimes tunneling rate is larger than radiative recombination rate, i.e., ⌫tunnel ⬎ ⌫, leading to broadening in the linewidths observed in photoluminescence measurements.48 Then from a Wentzel-Kramers-Brillouin estimation of the two different tunneling barriers together with the measured tunneling rate, we estimate the tunneling rate ⌫tunnel to be on the order of 0.02– 0.1 ns−1 in our structure. We take it to be independent of the gate voltage within the single-electron regime. Figure 10 shows the calculated cotunneling rate obtained with ex-

It is known that spin relaxation in higher-dimensional systems is mainly due to SO interaction in conjunction with phonons.50,51 Despite being strongly suppressed, SO interaction is still an enabling mechanism for phonon-assisted spin flips in QDs and a considerable amount of theoretical work has been done on this spin-relaxation mechanism.28,51–54 SO coupling is a well-known phenomenon in atomic physics as well as in semiconductors and is, in general, characterized by an interaction term of type HSO = 兺i,jaijˆli␴ˆ j, with ˆl the angular ˆ the spin operator of the electron; momentum operator and ␴ the sum runs over all pairs i , j = x, y, and z. The resulting spin-relaxation rate is a function of magnetic field and is given by

␬phonon =

共ge␮BBz兲5 ⌳p , ប共ប␻0兲4

共A7兲

where ប␻0 is the quantization energy for electrons and ⌳ p a dimensionless constant describing the strength of the piezoelectric coupling. The Bz5 dependence valid for electronic Zeeman splitting EZ,e Ⰷ kT becomes replaced by Bz4kT when EZ,e Ⰶ kT, due to the Boltzmann factor in Eq. 共C1兲.51,52 In addition to this dominant mechanism, there are numerous other ways of direct spin-phonon coupling which turn out to be orders of magnitude weaker than the admixture mechanism described above51 and are not considered in our treatment. Likewise, two-phonon processes with characteristically strong temperature dependence28,53 共T7 – T11兲 are also

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DREISER et al.

rather weak and not included here. Finally, it has been proposed that phonons together with the hyperfine-induced mixing of the Zeeman s levels lead to relaxation of the QD spin. As already mentioned in Sec. II, this mechanism is also inefficient and the resulting rate is predicted to depend on the external magnetic field as ⬃Bz3 共Ref. 29兲: according to the calculations presented in Ref. 27, the rate will be less than ␬ ⬃ 1 s−1 at a magnetic field of 1 T when considering the larger quantization energy in our QDs. Based on these arguments, we proceed with considering only the mechanism leading to Eq. 共A7兲. APPENDIX B: SAMPLE AND EXPERIMENTAL TECHNIQUES

Our InAs/ GaAs QDs are grown by molecular beam epitaxy in Stranski-Krastanow mode, leading to lens-shaped dots of average size 25⫻ 25⫻ 5 nm3; QD light emission is blueshifted by partially covered islands technique. A 35 nm GaAs tunneling barrier separates the QDs from a charge reservoir formed by a heavily doped n-GaAs layer which forms the back contact. Above the QDs, there is a 12-nm-thick GaAs cap and a 50-nm-thick Al0.4Ga0.6As blocking layer which prevents the holes from coupling to the continuum states within the 88 nm capping layer.47 Bias voltage between the back contact and a semitransparent 5 nm TiSchottky window determines the electric field in the structure and allows us to load a single conduction-band electron into the QD. All experiments described here are carried out with a confocal microscopy setup immersed in a liquid helium bath cryostat at a temperature of 4.2 K. The numerical aperture of the microscope is 0.68, resulting in a diffraction limited spot size of ⬃1 ␮m. Area density of QDs in our sample is low enough to have ⱕ3 dots in the focal spot simultaneously. Transmitted light is collected and sent to a circular polarization analyzer which distributes the light to two photodetectors, similar to that of Ref. 11. The initial step of our experiment is a gate sweep, i.e., a PL measurement as a function of gate voltage as shown in Fig. 11共a兲. For this we send in a laser, exciting electrons and holes in the bulk GaAs at an energy of ⬇1.6 eV. QD luminescence is sent to a grating spectrometer with a resolution ⬃20 ␮eV. Hereafter, the differently charged excitonic complexes can be identified by their characteristic emission energy and voltage dependence profile.55 From then on, we only apply resonant excitation of the QD single-electron ground-state transitions by using a DT technique.23–26 In order to obtain a spectrum, we sweep a single-mode Ti:sapphire laser over the QD transition and record the intensity of the transmitted light. A QD resonance is observed as a dip on top of the laser background 关see Fig. 11共b兲兴. The resolution of this technique is only limited by the laser linewidth, i.e., ⌬␯laser ⬍ 1 MHz. APPENDIX C:OPTICAL BLOCH EQUATIONS FOR THE THREE-LEVEL SYSTEM

FIG. 11. 共Color online兲共a兲 Example gate sweep. This plot has been obtained by increasing the gate voltage step by step and for each step taking a single-QD photoluminescence 共PL兲 spectrum. The three strongest emission lines are identified as X0, X1−, and tentatively X1+, which result from s-shell electron-hole recombination from differently charged excitonic complexes. Small continuous PL energy shift is due to quantum-confined stark shift. 共b兲 Example differential transmission laser scan. On resonance, Rayleighscattered light interferes with the laser background and results in a dip in the intensity measurement. The FWHM of the Lorentzian fit indicated by the solid red line is 460 MHz.

ˆ ˆ =H Zeeman + Hint,rad describes the unitary dynamics and Lˆrelaxation results from the interactions with reservoirs. After adding the relaxation terms due to the coupling to the thermal bath of radiation field modes 共spontaneous emission terms兲 at rate ⌫, the relaxation terms in the Lindblad form are56 ⌫ Lˆrelaxation = 共2␴ˆ 23␳ˆ ␴ˆ 32 − ␴ˆ 33␳ˆ − ␳ˆ ␴ˆ 33兲 + 2

␬ ¯n共2␴ˆ 12␳ˆ ␴ˆ 21 − ␴ˆ 22␳ˆ 2

␬ ¯ + 1兲共2␴ˆ 21␳ˆ ␴ˆ 12 − ␴ˆ 11␳ˆ − ␳ˆ ␴ˆ 11兲. − ␳ˆ ␴ˆ 22兲 + 共n 2 共C1兲 Here, ␴ˆ ab = 兩a典具b兩 is the projection operator. At temperatures smaller or comparable to the electronic Zeeman splitting kT ⬍ EZ,e, a Boltzmann factor ¯n = 1 / 关exp共ge␮BB / kT兲 − 1兴 needs to be taken into account, which leads to thermalization of the electron spin, i.e., in the absence of light, ␳11 / ␳22 = exp共−EZ,e / kT兲, where EZ,e is the electronic Zeeman energy. In the case of exchange coupling, the ¯n terms cannot be regarded as an occupancy; it can, however, be shown that a similar factor appears in the cotunneling rate 共A6兲 when Zeeman splitting is taken into account.57 The optical Bloch equations are derived from the master equation given at the beginning of this paragraph. Including rotating-wave approximation and taking the limit of Zeeman splitting Ez Ⰶ kT which eliminates the Boltzmann factors, the optical Bloch equations read

The master equation for the system 共reduced兲 density opˆ , ␳ˆ 兴 + Lˆ ˆ erator ␳ˆ reads dtd ␳ˆ = iប1 关H 0 relaxation, where the term H0 075317-12

d ␳11 = i⍀H共␳12 − ␳21兲 + ␥hm␳33 − ␬共␳11 − ␳22兲, dt



⍀R d ⬘ − ␳32 ⬘ 兲 + i⍀H共␳21 − ␳12兲 + ⌫␳33 + ␬共␳11 ␳22 = i 共␳23 2 dt

The last term describes coherence induced by the spontaneous relaxation into a superposition of dressed-basis ground states at a rate proportional to the occupation of the excited state ˜␳33. When multiplying with 具2兩 from the left and 兩1典 from the right, we obtain dtd ˜␳21 = −2␾⌫˜␳33. The same relation is obtained for ˜␳21 when multiplying with 具1兩 and 兩2典, respectively. The transformed ␬ terms keep the Lindblad form, and we obtain for EZ,e Ⰶ kT

− ␳22兲, ⍀R d ⬘ − ␳23 ⬘ 兲 − 共⌫ + ␥hm兲␳33 , ␳33 = i 共␳32 2 dt ⍀R d ⬘ + i⍀H共␳11 − ␳22兲 − ␬␳12 , ␳12 = i ␳13 2 dt

冉

˜␬ ˜ ˜ ˜L ˜ ˜␳ − ˜␳ M ˜ 兲 + 共2N ˜ ˜␳N ˜ ␳N12 − M relaxation,␬ = 关共2N21˜ 11 11 12 21 2

冊

⌫ + ␥hm + ␬ ⍀ d ⬘ = i R ␳12 − i⍀H␳23 ⬘ + − ⬘, ␳13 − i␦␻ ␳13 2 dt 2

冉

˜ ˜␳ − ˜␳ M ˜ 兲兴, −M 22 22

冊

⌫ + ␥hm + ␬ ⍀ d ⬘ = i R 共␳22 − ␳33兲 − i⍀H␳13 ⬘ + − ⬘, − i␦␻ ␳23 ␳23 2 dt 2 with

⬘e ␳13 = ␳13

i␻Lt

,

⬘e ␳23 = ␳23

i␻Lt

.

⬘*, ␳21 ⬘*, ␳32 ⬘*, and ␳11 + ␳22 + ␳33 = 1. We have ␳31 ⬘ = ␳13 ⬘ = ␳12 ⬘ = ␳23 APPENDIX D: DRESSED-STATE TRANSFORMATION

The transformation used to diagonalize the coupling to the ˜ quasistatic nuclear 共Overhauser兲 field can be written as H † † † ˆ = SHS and ˜␳ = S␳S , with ␾ = ⍀H / ␻z and S S = I. We assume ␾ Ⰶ 1. When taking only first-order terms

冢

1 −␾ 0

S= ␾ 0

1 0

冣

0 . 1

共D1兲

The spontaneous emission terms then yield SLrelaxation,⌫S† = ⌫2 共2S␴23S†˜␳S␴32S† − S␴33S†˜␳ − ˜␳S␴33S†兲. For the new projection operator S␴23S†, we obtain S␴23S† = ␾␴˜1˜3 + ␴˜2˜3 and the conjugate relation. Here, ␴˜i˜j = 兩i˜典具j˜兩. Using this with the ˜ previous relation, we obtain SLrelaxation,⌫S† = ␥2 共2␴˜1˜3˜␳␴˜3˜1

˜ = S ␴ S †, N ˜ =N ˜† , M ˜ = S␴ S†, and ˜␬ = ␬. with N 21 21 12 ii ii 21 APPENDIX E: NUMERICAL STUDIES

The derived formalism considers a static randomly oriented nuclear field. Within a measurement time, BN changes ⬃100 times. In order to calculate measurable quantities such as linewidths and peak heights as functions of electric and magnetic fields, we have, thus, performed numerical simulations: For a given set of parameters, the steady-state solutions of the optical Bloch equations as given in Appendix C are numerically evaluated, in particular Im(␳23共⬁兲) is then linked to the absorption 共details in Sec. III兲. A fluctuating hyperfine field is implemented by pulling three random numbers BN,i following Eq. 共A1兲. From BN,xy, using Eq. 共A3兲, state-mixing strength ⍀H 共A4兲 and pure Zeeman splitting ␻z are calculated before evaluating the density matrix steady state. This procedure is repeated in order to average over ⬃100 random settings of the hyperfine field. In the cases, the simulation could not be performed throughout the whole parameter space; we confirmed in key regimes that results agree well with that of a static Overhauser field with equal magnitude in x, y, and z: From Eq. 共A4兲, we obtain for the rms value of ⍀H共t兲

˜

⌫ − ␴˜3˜3˜␳ − ˜␳␴˜3˜3兲 + ⌫2 共2␴˜2˜3˜␳␴˜3˜2 − ␴˜3˜3˜␳ − ˜␳␴˜3˜3兲 − 2␾ 2 共␴˜2˜3˜␳␴˜3˜1 + ␴˜1˜3˜␳␴˜3˜2兲, where ˜␥ = ␾2⌫ and ˜⌫ = ⌫. ˜␴ij = 兩i˜典具j˜兩 is the projection operator acting on ˜␳. The first, ˜␥ term, corresponds to relaxation via a weak optical transition induced by the hyperfine field, allowing for spin-flip Raman events, and the second, ˜⌫ term, describes relaxation via the strong optical trion transition.

1 J.

Elzerman, R. Hanson, L. Van Beveren, B. Witkamp, L. Vandersypen, and A. Kouwenhoven, Nature 共London兲 430, 431 共2004兲. 2 S. Amasha, K. MacLean, I. Radu, D. M. Zumbühl, M. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys. Rev. Lett. 100, 046803 共2008兲. 3 F. Koppens, C. Buizert, K. Tielrooij, I. Vink, K. Nowack, T. Meunier, L. Kouwenhoven, and A. Vandersypen, Nature 共London兲

共D2兲

2 共t兲典 = 具⍀H

冉冊 g e␮ B ប

2

冉冊

具B2xy共t兲典 g e␮ B = 4 ប

2

2 Bnuc . 2

共E1兲

Here, the assumption for the observed absorption ⌰ to be made is 具⌰(B2N,xy共t兲)典 ⬇ ⌰(具B2N,xy共t兲典), i.e., the averaging over the absorption strength for different settings of the hyperfine field approximately equals the strength of absorption for the average field magnitude, equal in x, y, and z.

442, 766 共2006兲. F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, Science 309, 1346 共2005兲. 5 A. Johnson, J. Petta, J. Taylor, A. Yacoby, M. Lukin, C. Marcus, M. Hanson, and A. Gossard, Nature 共London兲 435, 925 共2005兲. 4

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R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 共2005兲. 7 M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. Finley, Nature 共London兲 432, 81 共2004兲. 8 D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 共1998兲. 9 A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. 83, 4204 共1999兲. 10 T. Calarco, A. Datta, P. Fedichev, E. Pazy, and P. Zoller, Phys. Rev. A 68, 012310 共2003兲. 11 M. Atatüre, J. Dreiser, A. Badolato, A. Högele, K. Karrai, and A. Imamoglu, Science 312, 551 共2006兲. 12 I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. B 65, 205309 共2002兲. 13 J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, Phys. Rev. B 76, 035315 共2007兲. 14 From the measured light polarization in our experiment, we obtain that the unwanted ␴− component is suppressed by a factorof 25. An additional factor comes from the optical detuning with respect to the diagonal transition in magnetic field, with the laser being on the strong transition. At Bz = 60 mT, the detuning leads to a further suppression ⬃1 / 50, altogether suppressing the unwanted excitation of the weak transition by more than 1200 times, increasing with magnetic field. At fields less or on the order of the hyperfine field, spin ground states are strongly mixed, leading to all possible 共vertical as well as diagonal兲 couplings in the four-level scheme, thus creating two differently polarized ⌳ systems with equal decay rates to both ground states. The presence of the 共unwanted and mainly suppressed兲 ␴− polarized light would just increase the bidirectional OSP by a small amount. 15 J. Brossel, A. Kastler, and J. Winter, Science 13, 668 共1952兲. 16 A. Shabaev, A. L. Efros, D. Gammon, and I. A. Merkulov, Phys. Rev. B 68, 201305共R兲共2003兲. 17 R. Loudon, The Quantum Theory of Light, 3rd ed. 共Oxford Science, New York, 2003兲. 18 J. M. Luttinger, Phys. Rev. 102, 1030 共1956兲. 19 G. Bester, S. Nair, and A. Zunger, Phys. Rev. B 67, 161306共R兲共2003兲. 20 This can be seen from 具↓ ↑ ⇑ hmix兩Hint,rad兩 ↓ 典 ⫽ 0 as opposed to 具↓ ↑ ⇑ 兩Hint,rad兩 ↓ 典 = 0. 21 K. Karrai and R. J. Warburton, Superlattices Microstruct. 33, 311 共2003兲. 22 A. Högele, Ph.D. thesis, LMU Munich, 2005. 23 B. Alen, A. Högele, M. Kroner, S. Seidl, K. Karrai, R. J. Warburton, A. Badolato, G. Medeiros-Ribeiro, and P. M. Petroff, Appl. Phys. Lett. 89, 123124 共2006兲. 24 B. Alen, F. Bickel, K. Karrai, R. J. Warburton, and P. M. Petroff, Appl. Phys. Lett. 83, 2235 共2003兲. 25 A. Högele, S. Seidl, M. Kroner, K. Karrai, R. J. Warburton, B. D. Gerardot, and P. M. Petroff, Phys. Rev. Lett. 93, 217401 共2004兲. 26 A. Högele, B. Alen, F. Bickel, R. J. Warburton, P. M. Petroff, and K. Karrai, Physica E 共Amsterdam兲 21, 175 共2004兲. 27 R. G. Newton, Am. J. Phys. 44, 639 共1976兲. 28 A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 64, 125316 共2001兲. 29 S. I. Erlingsson and Y. V. Nazarov, Phys. Rev. B 66, 155327 共2002兲. 30 P. A. Dalgarno, J. McFarlane, B. D. Gerardot, R. J. Warburton, K.

Karrai, A. Badolato, and P. M. Petroff, Appl. Phys. Lett. 89, 043107 共2006兲. 31 Due to our square-wave modulation scheme, inducing a nonexponential spin decay, the conversion from the known modulation frequency to ␬exp rate does not need to include a factor of 2␲. This has been confirmed by comparing the result of a numerical simulation using square-wave-shaped spin relaxation to the outcome of a simple rate equation model including exponential decay of spin. 32 C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu, Phys. Rev. Lett. 96, 167403 共2006兲. 33 B. Eble, O. Krebs, A. Lemaitre, K. Kowalik, A. Kudelski, P. Voisin, B. Urbaszek, X. Marie, and T. Amand, Phys. Rev. B 74, 081306共R兲共2006兲. 34 As a Gedanken experiment, we assume transfer of polarization from light to nuclei via the electron spin under a static positive magnetic field. Then, due to negative electron g factor, the spindown state has higher energy than the spin-up state. The laser is resonant with the red 共first case兲 or the blue 共second case兲 Zeeman transition. In the first case, electron spin is pumped to the spin-up state, which leads to nuclear spin-up polarization after an electron-nuclei flip-flop interaction. As the hyperfine constant A is positive, the energy of the electron spin-down state is consequently lowered, while the trion states remain untouched, leading to a blueshift of the red Zeeman optical transition which is being observed. The same happens in the second case, where spin-down nuclear polarization is created and the energy of the spin-up state is lowered, again leading to a blueshift of the observed transition. This scenario is clearly opposite of what we observe, thus ruling out the presence of an efficient DNSP. 35 D. V. Bulaev and D. Loss, Phys. Rev. Lett. 95, 076805 共2005兲. 36 D. Paget, G. Lampel, B. Sapoval, and V. I. Safarov, Phys. Rev. B 15, 5780 共1977兲. 37 P. Maletinsky, A. Badolato, and A. Imamoglu, Phys. Rev. Lett. 99, 056804 共2007兲. 38 b2 / N = 1 共 ␯0 兲2兺共A 兲2兩␺共R 兲兩4具I2典 = 1 关A2I共I + 1兲兴 / 共g ␮ 兲2; this i i i e B 0 i 3 8ge␮B 3N result is obtained by assuming a boxlike confinement potential, further transforming the sum into an integral and using the localization volume of the QD electron given by VL−1 = 兰dr兩␺共Ri兲兩4 = 8 / N␯0. 39 P. Maletinsky, C. W. Lai, A. Badolato, and A. Imamoglu, Phys. Rev. B 75, 035409 共2007兲. 40 In this context, we note that other experiments suggest that the dynamics of the nuclear spin ensemble could be altered by the measurement itself and the hyperfine field stays locked for a time on the order of seconds 共Ref. 4兲关Fig. 4共b兲兴. In our measurements, there is some evidence of alteration of dynamics at large magnetic fields and large gate-voltage detunings 共not shown in this work兲; however, at low magnetic fields, these locking effects do not seem to play a dominant role. 41 T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and S. Tarucha, Nature 共London兲 419, 278 共2002兲. 42 D. V. Averin and Y. V. Nazarov, Phys. Rev. Lett. 65, 2446 共1990兲. 43 D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. AbuschMagder, U. Meirav, and M. A. Kastner, Nature 共London兲 391, 156 共1998兲. 44 S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, 540 共1998兲. 45 A. O. Govorov, K. Karrai, and R. J. Warburton, Phys. Rev. B 67, 241307共R兲共2003兲.

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OPTICAL INVESTIGATIONS OF QUANTUM DOT SPIN… 46 R.

W. Helmes, M. Sindel, L. Borda, and J. von Delft, Phys. Rev. B 72, 125301 共2005兲. 47 S. Seidl, M. Kroner, P. A. Dalgarno, A. Högele, J. M. Smith, M. Ediger, B. D. Gerardot, J. M. Garcia, P. M. Petroff, K. Karrai, and R. J. Warburton, Phys. Rev. B 72, 195339 共2005兲. 48 J. M. Smith, P. A. Dalgarno, R. J. Warburton, A. O. Govorov, K. Karrai, B. D. Gerardot, and P. M. Petroff, Phys. Rev. Lett. 94, 197402 共2005兲. 49 With Zeeman splitting ␬↑→↓ ⫽ ␬↓→↑, i.e., the rate flipping the spin down is different from the rate flipping it up. The term f共␧兲关1 − f共␧兲兴 has to be replaced by f共␧ ⫿ ប␻z / 2兲关1 − f共␧ ⫾ ប␻z / 2兲兴 for ␬↓→↑ and ␬↑→↓, respectively. 50 M. I. D’yakonov, V. A. Marushchak, V. I. Perel’, and A. N. Titkov, Sov. Phys. JETP 63, 655 共1986兲. 51 A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B 61, 12639 共2000兲. 52 L. M. Woods, T. L. Reinecke, and Y. Lyanda-Geller, Phys. Rev. B 66, 161318共R兲共2002兲.

53 V.

N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 共2004兲. 54 D. V. Bulaev and D. Loss, Phys. Rev. B 71, 205324 共2005兲. 55 R. J. Warburton, C. Schaflein, D. Haft, F. Bickel, A. Lorke, K. Karrai, J. M. Garcia, W. Schoenfeld, and P. M. Petroff, Nature 共London兲 405, 926 共2000兲. 56 Y. Yamamoto and A. Imamoglu, Mesoscopic Quantum Optics 共Wiley, New York, 1999兲. 57 Including Zeeman splitting, relation 共A6兲 can be written in = exp共−ប␻ / 2kT兲兰 d␧A共␧兲f˜共␧兲 and, similarly, the form ␬

075317-15

↑→↓

z

␧

˜f 共␧兲 = 关兵1 + exp关␧ ␬↓→↑ = exp共ប␻z / 2kT兲兰␧d␧A共␧兲f˜共␧兲, with + 共ប␻z / 2kT兲兴其关1 + exp关␧ − 共ប␻z / 2kT兲兴其兴−1. In Eq. 共C1兲, the ¯ + 1兲 is ␬¯n term has to be replaced by ␬↑→↓, whereas the ␬共n replaced by ␬↓→↑. As a consequence, the ratio of spin-up versus spin-down state occupation is governed, as one would expect, by the Boltzmann factor ␬↑→↓ / ␬↓→↑ = exp共−ប␻z / kT兲.